Notational Conventions

This section provides the notational conventions used throughout this thesis.

A digital image ${g}=\{{g}_i: i \in \mathbf{R}\}$ on a 2D finite arithmetic lattice $\mathbf{R}=\{(x,y):0\leqslant x \leqslant \mathbf{M}-1;0\leqslant y \leqslant \mathbf{N}-1\}$ is a function $f: \mathbf{R}\rightarrow \mathbf{Q}$ that maps the supporting lattice $\mathbf{R}$ onto a finite set $\mathbf{Q}=\{0,1,...Q-1\}$ of signal values (e.g., grey levels, colours, or multi-band signatures). For simplicity, the set $\mathbf{Q}$ is assumed to be integer-valued in the range between 0 and $Q-1$.

A pixel in an image ${g}$ is denoted by ${g}_i$ where $i$ is the positioning index in the lattice $\mathbf{R}$. The set of all pixels in an image ${g}$ with exception of a pixel ${g}_i$ is denoted by ${g}^i$. Similarly, all sites on lattice $\mathbf{R}$ except the $i$th site is denoted by $\mathbf{R}^i$, so that ${g}^i=\{{g}_j :j \in \mathbf{R}^i= \mathbf{R}-\{i\} \}$.

The notation $\mathcal{N}_i$ defines a neighbourhood of a site $i$, which usually consists of a set of sites with a specific spatial configuration. All sites in a neighbourhood are dependent of each other.

Because the thesis deals mainly with probability texture models, in particular Markov-Gibbs random fields, related terminology and notations are presented below.

A random variable has a nondeterministic value with a given probability distribution. A sequence or array of statistically interrelated random variables form a discrete stochastic process. Let a random variable $\mathbf{S}_{i}\in\mathbf{Q}$ be associated with each site $i \in \mathbf{R}$. Then these variables form a random field  $\mathbf{S}=\{\mathbf{S}_i: i\in\mathbf{R};\mathbf{S}_{i}\in\mathbf{Q}\}$ in a configuration space  $\mathcal{S}$ if a probability measure $Pr(\mathbf{S}:\forall \mathbf{S}\in \mathcal{S})>0$ exists. A random field is a two-dimensional stochastic process. The configuration space has a combinatorial number of different images, $\vert\mathcal{S}\vert = {Q}^{\vert\mathbf{R}\vert}$.

Under a random field model, an image ${g}$ is considered as an instance or a sample of the configuration space $\mathcal{S}$. The image probability $Pr(S={g})$, or simply, $Pr({g})$, is known as the joint probability of the image signals. Another commonly-used probability measure is conditional probability of a pixel ${g}_i$ given all other pixels ${g}^i$, denoted by $Pr({g}_i \mid {g}^i)$. The conditional probability is a local probability of image signals, while the joint probability is a global one.

For convenience's sake, the notational conventions and symbols used in this thesis are summarised in Table 4.1.

Table 4.1: Notational conventions and symbols.

Notation	Meaning
$\mathbf{R}$	A 2D finite arithmetic lattice
$\mathbf{Q}$	The set of integral signal values
${g}$	A digital image defined on $\mathbf{R}$
${g}^i$	All pixels in the image ${g}$ except the pixel ${g}^i$
$\mathbf{S}$	A random field
$\mathcal{S}$	The configuration space of a random field $\mathbf{S}$
$\mathcal{N}_i$	Neighbourhood of the site $i$
$\mathbf{R}^i$	The lattice $\mathbf{R}$ with exception of the $i$th site
$\mathbf{S}^i$	All sites of $\mathbf{S}$ with exception of a site $S_i$
$Pr({g})$	Joint probability of a sample image ${g}$
$Pr({g}_i \mid {g}^i)$	Conditional probability of image signal $g_i$ at pixel $i$ given ${g}^i$